Groups and Group Actions: Lecture 8

In which we prove Lagrange’s theorem, and deduce many interesting results as a consequence.

• Lemma 33 (Coset equality test): Let $H$ be a subgroup of a group $G$.  Take $g_1$, $g_2 \in G$.  Have $g_1 H = g_2 H$ if and only if $g_2^{-1} g_1 \in H$.  For one direction we noted that $g_1 \in g_1 H$ so if the two left cosets are the same then $g_1 \in g_2 H$ and that gave us $g_2^{-1} g_1 \in H$.  For the other direction, we showed that $g_1 H \subseteq g_2 H$ and similarly the other way round.
• Theorem 34 (Lagrange‘s Theorem): Let $G$ be a finite group and let $H$ be a subgroup of $G$.  Then $|H| \mid |G|$.  We showed that the left cosets of $H$ partition $G$, and that each left coset of $H$ has the same size as $H$.  These two immediately give $|G| = |G/H| \times |H|$.
• Lemma 35: Let $G$ be a finite group.  Take $g \in G$.  Then $g$ has finite order. We noted that $g$, $g^2$, $g^3$, … all lie in the finite set $G$, so we must have some repetition $g^r = g^s$, from which the result followed.
• Corollary 36: Let $G$ be a finite group.  Take $g \in G$.  Then $o(g)$ divides $|G|$.  We noted that $\langle g \rangle$ is a subgroup of $G$ with order $o(g)$.
• Corollary 37: Let $p$ be prime, let $G$ be a finite group with order $p$.  Then $G$ is cyclic.  If $g \in G \setminus \{ e \}$, then we saw that $o(g) = p$, so $g$ is a generator of $G$.
• Corollary 38: Let $G$ be a finite group, take $g \in G$.  Then $g^{|G|} = e$. This follows from the facts that $o(g) \mid |G|$ and $g^{o(g)} = e$.
• Theorem 39 (Fermat‘s Little Theorem): Let $p$ be prime.  Let $a$ be an integer coprime to $p$.  Then $a^{p-1} \equiv 1 \pmod{p}$.  This is immediate from Corollary 38, since $(\mathbb{Z}_p^{\times}, \times)$ is a group of order $p-1$.
• Theorem 40 (Fermat-Euler Theorem): Let $n \geq 2$ be an integer.  Define $\phi(n) := |\{ k \in \mathbb{N} : 1 \leq k \leq n, \mathrm{hcf}(k,n)=1\}|$.  Let $a$ be an integer coprime to $n$.  Then $a^{\phi(n)} \equiv 1 \pmod{n}$.  This is another consequence of Corollary 38, this time noting that $(\mathbb{Z}_n^{\times}, \times)$ is a group of order $\phi(n)$.

Understanding today’s lecture

Can you relate Lemma 33 to the two examples of left cosets we saw at the end of the last lecture?  Check that it works out.

Having proved Lagrange’s Theorem, I mentioned that there is an alternative way to prove that the left cosets of $H$ partition $G$ using equivalence relations.  Define a relation $\sim$ on $G$ by $g_1 \sim g_2$ if and only if $g_2^{-1} g_1 \in H$.  Can you show that this is an equivalence relation?  What are the equivalence classes?  Now complete the proof.

Why is Fermat’s Little Theorem a special case of the Fermat-Euler Theorem?

When I reminded you of what an equivalence relation is, I gave as an example the relation $\sim$ on a group $G$, defined by $g_1 \sim g_2$ if and only if $g_1 = g_2$ or $g_1 = g_2^{-1}$.  Can you show that this really is an equivalence relation?  What are the equivalence classes?

Let $p$ be a prime.  Which elements of $\mathbb{Z}_p^{\times}$ (a group under multiplication) are self-inverse (have $g = g^{-1}$)?  What are the equivalence classes for the equivalence relation $\sim$ from the last paragraph when applied to this group?  What does this tell us about the product $(p-1)!$ modulo $p$?  The result here is called Wilson‘s theorem; you can read more about it in Richard Earl’s online notes, or in my blog post.

Let $G$ be a finite group with even order.  Can you show that $G$ must contain an element of order 2?  (Hint: use the equivalence relation from the paragraph before last.)  Again, you can read more about this in the online notes, but I strongly recommend that you try to deduce it for yourself first, it’s good practice.

Further reading

I’ve written about several of these topics before on this blog.  I’ve written about Lagrange’s Theorem (not be confused with another Lagrange’s Theorem that I’ve also written about), and about Fermat’s Little Theorem (that post outlines the three proofs I mentioned in today’s lecture).  You’ll find a handy video with our statement and proof of Lagrange’s Theorem at hedgehogmaths.  Remember to watch out for tigers.

The Euler totient function has many interesting properties.  Can you show that if $p$ is prime then $\phi(p) = p-1$?  What is $\phi(p^k)$ for a general integer $k \geq 1$?  It turns out that $\phi$ is multiplicative: if $m$ and $n$ are coprime, then $\phi(mn) = \phi(m) \phi(n)$.  (Note that it’s important here that $m$ and $n$ are coprime!)  Can you prove this?  There’s a nice connection with the Chinese Remainder Theorem.  You can read more about the function in a number theory book (such as The Higher Arithmetic by Davenport), or online (eg of course Wikipedia has a page).

Here’s some information from Wikipedia about the RSA cryptosystem that I mentioned at the end of the lecture as using Fermat-Euler.

Preparation for Lecture 9

Lecture 9 is weeks away, so for now I suggest that you prepare by working on the material from the first four weeks of the course.  If you understand that really well, then you’ll be ready for next term.  I’ll put some more specific questions up for you to consider a few days before Lecture 9, so check back nearer the time.  Here’s one for you to ponder for now.

Let $G$ and $H$ be isomorphic groups, and let $\theta: G \to H$ be an isomorphism.  Take $g \in G$.  What can you say about the orders of $g$ in $G$ and $\theta(g)$ in $H$?  How are they related?

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4 Responses to “Groups and Group Actions: Lecture 8”

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